When you have eliminated the impossible, whatever remains,
however improbable, must be the truth.



Fig. 1  Sir Arthur Conan Doyle.



Long held to be an inviolable rule of logic, this is actually a
sentence often placed in the mouth of fictional detective Sherlock
Holmes by his creator Sir Arthur Conan Doyle (Fig. 1). At its core is
the straightforward process of elimination. And lately it has become
the watchcry of conspiracists who quite obviously want us to believe
in something improbable, and quite often attempt to elicit that belief
by appearing to "eliminate the impossible."
The statement is true enough under the right consequences. In
deductive logic we know not to reject a proposition for which a
suitable proof has been constructed, simply on the basis that it seems
absurd. A deductive case, if inferred according to sound rules of
inference from premises known to be correct, is airtight.
In mathematical reasoning the Holmesian Maxim is crucial.
An integer, for example, must either be positive, negative, or zero.
If we wish to prove some given value is negative and a direct proof is
problematic, but we can show that it is neither positive nor zero,
then we will have proved rigorously by indirection that it must
therefore be negative. This would hold even if a prior informal
assessment of the problem suggests otherwise. Very often we must rely
on proving what something is not, and this is a powerful tool
in categorical logic.
However, in inductive reasoning the Holmesian Maxim does
not provide a helpful strategy.
Recall that inductive reasoning starts with specific observations
and attempts to collate them to form a single, more general
conclusion. This is what happens in a court of law. Bits of evidence
are presented which, when considered collectively, help the court
arrive at a general declaration of guilt or innocence. Inductive
cases are never airtight the way deductive or mathematical proofs must
be. Induction does not assure correctness, and as such relies on a
certain allowance for error, known as "inductive skepticism".
Induction governs historical investigations, which covers the
explanation of all phenomena and the characterization of all events
whether recent or ancient. The same principles that govern an
investigation of how the Romans breached the fortress of Masada also
govern the identification of a UFO sighted yesterday: one begins with
observations of artifacts and accounts and then forms a picture of
what really happened. The historian has no delusion of being able to
do so conclusively or correctly. He merely strives to draw a
conclusion which among its peers is the most likely to explain
the given evidence material.
Therein lies the difficulty. In the mathematical example above,
we can perfectly enumerate the available possibilities: positive,
negative, and zero. Categorical reasoning by definition presents us
with such clearly partitioned categories. But this esoteric purity is
not available in inductive investigations. There we are confronted
with the reality that an empirical observation might quite literally
have been produced by almost anything. And we are also faced with the
possibility that the one true answer may lie beyond our ability to
imagine it, and therefore even to consider or test it.
Let us say that again: the true antecedent to any observation may
not occur to us during our investigation. The indirect method of
reasoning, expressed in the Holmesian Maxim, requires that we
specifically enumerate all the possibilities that may apply to
the observation  whether or not we favor them. And it requires us
to know conclusively that we have considered all of them. In
an artificial context such as mathematics, blessed with rigid
definitions, this assurance of completeness is easy to come by. Not
so in the "messy" real world.
Even if we could establish, by some good fortune, the assurity of
having enumerated all the possibilities in preparation for examining
them, we would still have the task of soundly disproving them. And in
a historical inquiry that means an empirical proof with limited
available evidence  in other words, a limited ability to draw
unassailable conclusions. Bear strongly in mind that an indirect
proof requires conclusive disproof of the competing hypotheses. If
the argument can show only that some competitor is merely improbable
and not fully impossible, then it fails the stipulation of the
Holmesian Maxim. The choice is then between two improbables, not the
assertion of an improbability over an impossibility. And then the
relative merits of each become important and one must supply direct
proof for the desired proposition.
Historical inquiries are limited to the evidence at hand. The
nature of empirical investigations does not allow us to develop
crucial evidence on a whim. And so it may happen that the bit of
evidence necessary to substantiate or eliminate a hypothesis is simply
not available and not likely to be forthcoming. In the absence of key
evidence we cannot rule either way on the hypothesis, and that
conspicuously disallows us from saying it is impossible. And this
means we cannot use an indirect proof.
Disproof often requires the proof of a negative proposition. And
proving a negative is almost always difficult. For example, to
disprove the hypothesis The shadow in that photograph was
cast by the sun. is equivalent to proving its converse,
The shadow in that photograph was not cast by the
sun. This requires the author to exhaustively examine all
the ways in which suncast shadows can be made to appear, and to
specifically eliminate that any of them could have been employed in
the photograph in question. That's a very tall order.
The practical impossibility of assuring a complete set of
competing hypotheses, together with the limitations on disproving them
constitute an imposing barrier to successful inductive proofs
constructed according to the Holmesian blueprint.
So why is this method so common among conspiracists?
First, it creates the illusion of support for a proposition that
has no direct evidence at all in favor of it. Since most
conspiracyrelated propositions are pure conjecture, a direct proof is
not possible. The conspiracist would have nothing to write at all if
not for the practice of indirect proof.
Second, an indirect proof carries a semblance of rigor. If the
author is unable to fully enumerate the competing hypotheses then it's
unlikely the sympathetic reader will be able to; and thus he won't
necessarily notice the absence of a serious competitor that the author
has failed to consider. Unless the reader is predisposed to dig
deeper than the author, he will likely consider the case complete and
decisive.
This circumstance is especially effective on specialized topics
such as science. Most readers are not experts in all technical
matters. And so competing hypotheses that arise from obscure  but
nevertheless valid  scientific principles are likely to be missed by
both author and reader. And when a critic brings up these obscure
principles in his objection to the conspiracy theory, he can often be
accused of trying to muddy what would otherwise be a "straightforward"
case.
In this devious way, the conspiracist pares down the set of
possibilities to something which appeals to the "common sense"
intuition of the reader. The reader is spoonfed just enough
information to validate an overly simplistic view of the problem. And
this simpleton view then generates what appears to be a small set of
competing hypotheses, which the author can usually convince the reader
is complete. Then after disqualifying each of this small set of straw
men as an explicator, the author proclaims his desired proposition
vindicated by process of elimination.
Third, the Holmesian Maxim supplies language to predispose the
reader to accept a conclusion he might otherwise reject as absurd.
The astute conspiracist author realizes that his controversial
proposition will encounter skepticism. By introducing his attempt at
indirect proof with the Holmesian Maxim, the author imparts a degree
of intellectual comfort to the reader who can then accept the
proposition against his better judgment. The reader believes himself
to have remained rational if he accepts a preposterous conclusion that
nevertheless must be true by the process of elimination.
While conspiracists can easily create indirect inductive proofs
that seem rigorous even when applied to baseless propositions, they
seldom acknowledge the ease with which such indirect proofs can be
refuted. The two impassable obstacles in an indirect inductive proof
 assurance of completeness and strength of elimination  give
predictable rise to the two basic methods of refutation.
Any plausible competing hypothesis that the author does not
consider in his indirect proof, is sufficient refutation of the proof.
It does not matter whether the competitor's proponent is able to prove
the competing hypothesis in the specific case. It matters only
whether the author is able to disprove it in the specific case.
The author has the burden of proof to "eliminate the impossible". The
critic's burden of proof is for mere plausibility  that it is "not
impossible". So saith the Holmesian Maxim.
Since each competitor must be conclusively eliminated, the
strength of each eliminative proof must be aggressively tested. As
noted above, the eliminations are, by nature, difficult proofs to
construct to sufficient rigor. And the lack of empirical evidence may
eliminate the testability altogether, in which case impossibility may
not be assumed. But very often the simplism of a putative
elimination is its own undoing; it may serve only to suggest that a
hypothesis is improbable, not that it is truly impossible. And as
stated above, this reduces the argument to an evaluation of relative
probability among improbable hypotheses.
To compare one hypothesis to another on the basis of its relative
probability is the process of direct inductive proof. One must
examine the merits of the desired hypothesis, not the conspicuous lack
of merit in all its competitors. And because an indirect inductive
proof invariably reduces, upon scrutiny, to a direct proof, the smart
proponent adopts a direct proof strategy at the outset. And knowing
that a purely conjectural hypothesis cannot prevail according to a
direct proof, the smart proponent avoids advancing a conjectural
hypothesis altogether. And this leaves the Holmesian Maxim safely
where it belongs  away from the inductive case.
